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Find the Cartesian equation of plane through a given point and with a given perpendicular line

We say that a line parallel to a vector N (non-zero) is perpendicular to a plane M if N is normal to M. Given that a plane M goes through the point (2,3,-7) and that the line through the points (1,2,3) and (2,4,12) is perpendicular to M find the Cartesian equation of M.


First, N = (2,4,12) - (1,2,3) = (1,2,9). Therefore, the Cartesian equation of M is of the form

    \[ x + 2y + 9z = d. \]

Since (2,3,-7) is on the plane we have d = 2 + 6 - 63 = -55. Thus, the Cartesian equation of M is

    \[ x + 2y + 9z + 55 = 0. \]

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